exclusive vector meson production with impact parameter

Exclusive vector meson production with impact parameter dependentBalitsky-Kovchegov evolution equation [1]

Dagmar Bendovain collaboration with M. Matas, J. Cepila and J. G. Contreras

Czech Technical University in PragueFaculty of Nuclear Sciences and Physical Engineering

19. Zimanyi School - Winter Workshop on Heavy Ion Physics, Budapest, Hungary

4th December, 2019

[1]Based on: Phys. Rev. D 99, 051502 and Phys. Rev. D 100, 054015Dagmar Bendova 1 / 13

Motivation: Saturation of gluon densities

Figure: Diagram picturing the QCD evolutionof the partonic structure of the proton.C. Marquet, Nucl.Phys. A904-905 (2013) 294c-301c.

Exclusive production of a vector meson

is very sensitive to gluon densities.

An excellent probe of saturation effects.

Figure: Exclusive production of a vector meson.

Dagmar Bendova 2 / 13

Motivation: Saturation of gluon densities

Figure: Diagram picturing the QCD evolutionof the partonic structure of the proton.C. Marquet, Nucl.Phys. A904-905 (2013) 294c-301c.

Exclusive production of a vector meson

is very sensitive to gluon densities.

An excellent probe of saturation effects.

Figure: Exclusive production of a vector meson.

Vector meson production in the color dipole picture

Exclusive cross section to produce the vector meson

dσγ∗p→VMp

∣∣∣∣T ,L

(RT ,Lg

)2(1 + β2

∣∣〈Aγ∗p→VMpT ,L 〉

∣∣2.

The scattering amplitude is given as a convolution of the photon-VM wave function

with the dipole cross section:

AT ,L(x ,Q2, ~∆) = i

∫d~r

∫d~b|Ψ∗VMΨγ∗ |T ,L exp

[−i(~b − (1− z)~r

)~∆] dσqq

γ∗VM

1− z

x x′

(1− z)r

Figure: Schematic picture of vector meson production within the color dipole approach.

Vector meson production in the color dipole picture

Exclusive cross section to produce the vector meson

dσγ∗p→VMp

∣∣∣∣T ,L

(RT ,Lg

)2(1 + β2

∣∣〈Aγ∗p→VMpT ,L 〉

∣∣2.The scattering amplitude is given as a convolution of the photon-VM wave function

with the dipole cross section:

AT ,L(x ,Q2, ~∆) = i

∫d~r

∫d~b|Ψ∗VMΨγ∗ |T ,L exp

[−i(~b − (1− z)~r

)~∆] dσqq

γ∗VM

1− z

x x′

(1− z)r

Figure: Schematic picture of vector meson production within the color dipole approach.

Dipole-target cross section

Carries information about the strong interaction between the dipole and the proton.

The differential dipole-proton cross section is given by the dipole scattering

amplitudedσqq

d~b= 2N(x , ~r , ~b).

Approach to the dipole amplitude N(x , ~r , ~b)

I Factorization of the impact parameter dependence [2]

d~b→ σ0N(x , ~r)Tp(~b).

F Tp(~b) – proton profile in the impact-parameter plane

F σ0 – multiplicative factor obtained from fit to data or from normalization

I Numerical solution of the Balitsky–Kovchegov (BK) equation with b-dependence.

F Demanding on computational resources.

F Until recently, the problem of Coulomb tails.

[2]e.g. Phys. Lett. B766 (2017) 186; Nucl. Phys. B934 (2018) 330; Phys. Rev. D99 (2019) 034025

amplitudedσqq

d~b= 2N(x , ~r , ~b).

d~b→ σ0N(x , ~r)Tp(~b).

amplitudedσqq

d~b= 2N(x , ~r , ~b).

d~b→ σ0N(x , ~r)Tp(~b).

amplitudedσqq

d~b= 2N(x , ~r , ~b).

d~b→ σ0N(x , ~r)Tp(~b).

Balitsky–Kovchegov evolution equation

Evolution with rapidity Y ∼ ln(

)of the scattering amplitude N(~r , ~b,Y ) of a qq

dipole with a hadronic target

∂N(~r , ~b,Y )

∫d~r1K(r , r1, r2)

(N(~r1, ~b1,Y ) + N(~r2, ~b2,Y ) − N(~r , ~b,Y ) − N(~r1, ~b1,Y )N(~r2, ~b2,Y )

Describes the dressing of a color dipole with the evolution to higher energies.

J. L. Albacete et al., Phys. Rev. D 71 (2005) 014003

Balitsky–Kovchegov evolution equation

Evolution with rapidity Y ∼ ln(

)of the scattering amplitude N(~r , ~b,Y ) of a qq

dipole with a hadronic target

∂N(~r , ~b,Y )

∫d~r1K(r , r1, r2)

(N(~r1, ~b1,Y ) + N(~r2, ~b2,Y ) − N(~r , ~b,Y ) − N(~r1, ~b1,Y )N(~r2, ~b2,Y )

Describes the dressing of a color dipole with the evolution to higher energies.

J. L. Albacete et al., Phys. Rev. D 71 (2005) 014003

Evolution kernel

Describes the probability of a gluon emission.

Different approximations of the probability calculation lead to several forms of

kernels.

Running coupling kernel [3]

K rc(r , r1, r2) =αS(r 2)NC

r 21 r

(αS(r 2

αS(r 22 )− 1

(αS(r 2

αS(r 21 )− 1

Collinearly improved kernel [4]

K ci (r , r1, r2) =αS

r 21 r

min(r 21 , r

]±αSA1

(√Lr1rLr2r

KDLA(ρ) =J1(2

√αSρ2)√αSρ

, Lri ,r = ln

[3]I. Balitsky, Phys. Rev. D 75 (2007) 014001[4]E. Iancu et al., Phys. Lett. B 750 (2015) 643; A. Sabio Vera, Nucl. Phys. B722 (2005) 65

Evolution kernel

kernels.

r 21 r

(αS(r 2

αS(r 22 )− 1

(αS(r 2

αS(r 21 )− 1

K ci (r , r1, r2) =αS

r 21 r

min(r 21 , r

]±αSA1

(√Lr1rLr2r

KDLA(ρ) =J1(2

√αSρ2)√αSρ

, Lri ,r = ln

[3]I. Balitsky, Phys. Rev. D 75 (2007) 014001[4]E. Iancu et al., Phys. Lett. B 750 (2015) 643; A. Sabio Vera, Nucl. Phys. B722 (2005) 65

Evolution kernel

kernels.

r 21 r

(αS(r 2

αS(r 22 )− 1

(αS(r 2

αS(r 21 )− 1

K ci (r , r1, r2) =αS

r 21 r

min(r 21 , r

]±αSA1

(√Lr1rLr2r

KDLA(ρ) =J1(2

√αSρ2)√αSρ

, Lri ,r = ln

)[3]I. Balitsky, Phys. Rev. D 75 (2007) 014001[4]E. Iancu et al., Phys. Lett. B 750 (2015) 643; A. Sabio Vera, Nucl. Phys. B722 (2005) 65

Implementation of impact parameter dependence

The equation is solved using a new initial condition that takes into account the

location of the end-points of the dipole

N(~r , ~b,Y = 0) = 1− exp

4r 2T (~bq1 ,

~bq2 )

]T (~bq1 ,

~bq2 ) = exp

(−b2q1

)+ exp

(−b2q2

proton

J. Cepila, J. G. Contreras, M. Matas; Phys. Rev. D 99, 051502

The r behavior mimics the models

for the b-independent dipole

amplitude.

The b behavior contains an

exponential fall-off for dipole-ends

far away from the target.

Implementation of impact parameter dependence

The equation is solved using a new initial condition that takes into account the

location of the end-points of the dipole

N(~r , ~b,Y = 0) = 1− exp

4r 2T (~bq1 ,

~bq2 )

]T (~bq1 ,

~bq2 ) = exp

(−b2q1

)+ exp

(−b2q2

proton

The r behavior mimics the models

for the b-independent dipole

amplitude.

The b behavior contains an

exponential fall-off for dipole-ends

far away from the target.

The problem of Coulomb tails

Evolution with exponentially falling initial condition and K rc increases large dipoles

into a power-like growth.

I Destroys the predictive power of BK equation, doesn’t allow for phenomenological

applications.

I The growth violates the Martin-Froissart bound.

10−1 100 101 102

b [GeV−1]

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Scattering amplitude for r = 1.0 [GeV−1]

Initial condition N(b, Y = 0)

Nci(b, Y = 1)

Nci(b, Y = 3)

Nci(b, Y = 10)

Nrc(b, Y = 1)

Nrc(b, Y = 3)

Nrc(b, Y = 10)

Evolution at high-b can be suppressed by

suppressing large daughter dipoles

I By introducing a kernel cut-off a

F cut-off too strong

F predictive power isn’t restored

I Using a collinearly improved kernel K ci

F ressums single and double collinear

logarithms

F large daughter dipoles are suppressed.

aJ. Berger, A. M. Stasto, Phys. Rev. D 84 (2011) 094022

The problem of Coulomb tails

Evolution with exponentially falling initial condition and K rc increases large dipoles

into a power-like growth.

I Destroys the predictive power of BK equation, doesn’t allow for phenomenological

applications.

I The growth violates the Martin-Froissart bound.

10−1 100 101 102

b [GeV−1]

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Scattering amplitude for r = 1.0 [GeV−1]

Initial condition N(b, Y = 0)

Nci(b, Y = 1)

Nci(b, Y = 3)

Nci(b, Y = 10)

Nrc(b, Y = 1)

Nrc(b, Y = 3)

Nrc(b, Y = 10)

Evolution at high-b can be suppressed by

suppressing large daughter dipoles

I By introducing a kernel cut-off a

F cut-off too strong

F predictive power isn’t restored

I Using a collinearly improved kernel K ci

F ressums single and double collinear

logarithms

F large daughter dipoles are suppressed.

aJ. Berger, A. M. Stasto, Phys. Rev. D 84 (2011) 094022

Phenomenology with b-BK

Impact parameter dependent dipole amplitude can be used to study several

processes and observables

I Deep Inelastic Scattering - structure functions F2

I Gluon distribution functions (Weiszacker–Williams distribution)

I Nuclear F2

I Exclusive photo- and electroproduction of vector mesons (this talk)

I Nuclear F2

Predictions and comparison with data for cross sections of J/ψ meson

Predictions compared to H1 and ALICE p–Pb data.

0 0.2 0.4 0.6 0.8 1 1.2

]2|t| [GeV

Photoproduction

H1 (2006), <W> = 100 GeV Model

H1 (2013), <W> = 78 GeV Model

H1 (2013), <W> = 55 GeV Model

0 0.2 0.4 0.6 0.8 1 1.2

]2|t| [GeV

1−10

= 100 GeVpγ

Electroproduction, W

H1 (2006) 2

= 3.2 GeV2

Model, Q

H1 (2006) 2

= 7.0 GeV2

Model, Q

H1 (2006) 2

= 22.4 GeV2

Model, Q

[GeV]pγW

Photoproduction

H1 (2006)

H1 (2013)

ALICE pPb (2014)

ALICE pPb (2018)2

= 0.05 GeV2

Model, Q

Electroproduction

2 = 3.2 GeV

2Model, Q

2 = 7 GeV

2Model, Q

2 = 22.4 GeV

2Model, Q

H1 (2006)

D. Bendova, J. Cepila, J. G. Contreras, M. Matas, Phys. Rev. D100 (2019) 054015; references to experimental data in backup

0 0.2 0.4 0.6 0.8 1 1.2

]2|t| [GeV

Photoproduction

H1 (2006), <W> = 100 GeV Model

H1 (2013), <W> = 78 GeV Model

H1 (2013), <W> = 55 GeV Model

0 0.2 0.4 0.6 0.8 1 1.2

]2|t| [GeV

1−10

= 100 GeVpγ

H1 (2006) 2

= 3.2 GeV2

Model, Q

H1 (2006) 2

= 7.0 GeV2

Model, Q

H1 (2006) 2

= 22.4 GeV2

Model, Q

[GeV]pγW

Photoproduction

H1 (2006)

H1 (2013)

ALICE pPb (2014)

ALICE pPb (2018)2

= 0.05 GeV2

Model, Q

Electroproduction

2 = 3.2 GeV

2Model, Q

2 = 7 GeV

2Model, Q

2 = 22.4 GeV

2Model, Q

H1 (2006)

0 0.2 0.4 0.6 0.8 1 1.2

]2|t| [GeV

Photoproduction

H1 (2006), <W> = 100 GeV Model

H1 (2013), <W> = 78 GeV Model

H1 (2013), <W> = 55 GeV Model

0 0.2 0.4 0.6 0.8 1 1.2

]2|t| [GeV

1−10

= 100 GeVpγ

H1 (2006) 2

= 3.2 GeV2

Model, Q

H1 (2006) 2

= 7.0 GeV2

Model, Q

H1 (2006) 2

= 22.4 GeV2

Model, Q

[GeV]pγW

Photoproduction

H1 (2006)

H1 (2013)

ALICE pPb (2014)

ALICE pPb (2018)2

= 0.05 GeV2

Model, Q

Electroproduction

2 = 3.2 GeV

2Model, Q

2 = 7 GeV

2Model, Q

2 = 22.4 GeV

2Model, Q

H1 (2006)

Predictions and comparison with data for cross sections of φ meson

Predictions compared to H1 and ZEUS data.

0 0.2 0.4 0.6 0.8 1

]2|t| [GeV

1−10

Electroproduction, <W> = 75 GeV

ZEUS (2005) 2

= 2.4 GeV2

ZEUS (2005) 2

= 3.6 GeV2

H1 (2010) 2

= 6.6 GeV2

ZEUS (2005) 2

= 12.6 GeV2

[GeV]pγW

310 [n

ZEUS (2005)

2 = 2.4 GeV2Model, Q

ZEUS (2005), H1 (2010)

ZEUS (2005)

H1 (2010)

2 = 6.6 GeV2Model, Q2 = 13 GeV2Model, Q

Predictions and comparison with data for cross sections of φ meson

Predictions compared to H1 and ZEUS data.

0 0.2 0.4 0.6 0.8 1

]2|t| [GeV

1−10

Electroproduction, <W> = 75 GeV

ZEUS (2005) 2

= 2.4 GeV2

ZEUS (2005) 2

= 3.6 GeV2

H1 (2010) 2

= 6.6 GeV2

ZEUS (2005) 2

= 12.6 GeV2

[GeV]pγW

310 [n

ZEUS (2005)

ZEUS (2005), H1 (2010)

ZEUS (2005)

H1 (2010)

2 = 6.6 GeV2Model, Q2 = 13 GeV2Model, Q

Predictions and comparison with data for cross sections of Υ meson

Photoproduction predictions compared to H1, ZEUS, CMS and LHCb data.

[GeV]pγW

Photoproduction data

ZEUS (2009)

H1 (2000)

LHCb (2015)

CMS (2019)

Conclusions

The collinearly improved kernel suppresses the Coulomb tails → allows for

phenomenological applications.

Successful application of dipole amplitudes obtained by solving b-BK into exclusive

vector meson production.

Good agreement of the predictions with data for φ, J/ψ, and Υ(1S) mesons.

Conclusions

BACKUP SLIDES

Wave functions

The overlap between the photon and vector meson wave functions:

|Ψ∗VMΨγ∗ |T = ef eNC

πz(1− z)

f K0(εr)φT (r , z)−(z2 + (1− z)2

)εK1(εr)∂rφT (r , z)

|Ψ∗VMΨγ∗ |L = ef eNC

π2Qz(1− z)K0(εr)

[MVMφL(r , z) + δ

m2f −∇2

MVMz(1− z)φL(r , z)

ε = z(1− z)Q2 + m2f ; r ≡ |~r |

Scalar part of VM wave function from Boosted Gaussian model:

φT ,L(r , z) = NT ,Lz(1− z) exp

[− m2

8z(1− z)− 2z(1− z)r 2

φ2ST ,L(r , z) = φT ,L(r , z)

(1 + α2S

4z(1− z)− 4z(1− z)r 2

R2−m2

Corrections

Correction on the real part of the amplitude

I AT ,L is not purely imaginary, has a real part

λT ,L ≡∂ ln

∗p→VMpT ,L

)∂ ln

) , βT ,L = tan

(πλT ,L

Skewedness correction

I Takes into account the fact, that there are two x involved in the dipole-target

interaction

Rg (λT ,L) =22λT,L+3

Γ(λT ,L + 5

)Γ(λT ,L + 4)

γ∗VM

1− z

x x′

(1− z)r

Influence of the individual collinear kernel terms on the Coulomb tails

K 1ci =

r 21 r

; K 2ci =

min(r 21 , r

]±αSA1

; K 3ci = KDLA

(√Lr1rLr2r

10−1 100 101 102b [GeV−1]

10−7

10−6

10−5

10−4

10−3

10−2

10−1

1.0 GeV

−1, b

K1ci with αs=0.2

K1ci with αs=αs(rmin)

K1ci *K2

ci with αs=αs(rmin)K1ci *K3

ci with αs=αs(rmin)KciKrc with αs=αrunInitial c nditi n N(b, Y = 0)

DIS: Structure function F2(x ,Q2)

Cross section for γ∗p scattering and structure function F2

σγ∗p

T ,L =∑f

∫d2r

∫dz |Ψ∗Ψ|fT ,Lσqq(x , r), F2(x ,Q2) =

4π2α(σT + σL).

10−6 10−5 10−4 10−3 10−2x

10−1

F 2(x, Q

Q2=2.0 GeV2 * 20Q2=2.7 GeV2 * 21Q2=3.5 GeV2 * 22Q2=4.5 GeV2 * 23Q2=6.5 GeV2 * 24Q2=8.5 GeV2 * 25Q2=10.0 GeV2 * 26Q2=12.0 GeV2 * 27Q2=15.0 GeV2 * 28Q2=22.0 GeV2 * 29Q2=35.0 GeV2 * 210Q2=60.0 GeV2 * 211Q2=90.0 GeV2 * 212

References to experimental data

J/ψ data

I A. Aktas et al. (H1 Collaboration), Eur. Phys. J. C 46 (2006) 585

I C. Alexa et al. (H1 Collaboration), Eur. Phys. J. C 73 (2013) 2466

I B. B. Abelev et al. (ALICE Collaboration), Phys. Rev. Lett. 113 (2014) 232504

I S. Acharya et al. (ALICE Collaboration), Eur. Phys. J. C 79 (2019) 402

φ data

I F. D. Aaron et al. (H1 Collaboration), J. High Energy Phys. 05 (2010) 032

I S. Chekanov et al. (ZEUS Collaboration), Nucl. Phys. B718 (2005) 3

Υ(1S) data

I C. Adloff et al. (H1 Collaboration), Phys. Lett. B 483 (2000) 23

I S. Chekanov et al. (ZEUS Collaboration), Phys. Lett. B 680 (2009) 4

I R. Aaij et al. (LHCb Collaboration), J. High Energy Phys. 09 (2015) 084

I A. M. Sirunyan et al. (CMS Collaboration), Eur. Phys. J. C 79 (2019) 277

exclusive vector meson production with impact parameter

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